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Big, Bigger, Biggest

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

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Complex Sine

Solve the equation sin z = 2 for complex z. You only need the formula you are given for sin z in terms of the exponential function, and to solve a quadratic equation and use the logarithmic function.

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Sierpinski Triangle

What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.

Equation Attack

Age 16 to 18 Challenge Level:

When you meet a class of equations like this it would be good to find a general formula involving $a$ and $b$ (like the formula for solutions of quadratic equations).

It often helps towards a general solution to try special cases as suggested in this problem and exact solutions can be found in special cases. More generally however there is no closed formula for solutions and you will need to find approximate solutions by numerical methods.

This problem involves the use of logs, solving a quadratic equation and using a numerical method such as interval halving or the Newton Raphson method. To speed up the process of solving this equation for a particular pair of values of $a$ and $b$ you can use a programmable calculator or a spreadsheet.